I am trying to find the solution for this polynomial system. For some context it represents a game where every player has to choose a location in the x-line, given that it is best to be closer to 0, but it is also good to be far from other players.
$$ \begin{cases} C- ax_1^2 + \sum_{i=1}^n (x_1-x_i)^2 =0 \\ C- ax_2^2 + \sum_{i=1}^n (x_2-x_i)^2 =0 \\ \vdots \\ C- ax_n^2 + \sum_{i=1}^n (x_n-x_i)^2 =0 \end{cases} $$
With $C$ and $a$ are constants, $n$ is the number of players, and $\{x_1,...,x_n\}$ is the real numbers I wish to solve for.
I have looked for materials on how to solve polynomial equations but I could not find an easily understandable one. I understand that Grobner basis, and a potential representation of this system throught the use of rings and companion matrices. But I have not understood how to apply these concepts in an actual problem. Any tips on how does one solve problems like this in general is would also be greatly appreciated.
I could use simple substitution but I would like to keep the problem as general as possible without fixing $n$. Any help would be appreciated